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Shortcut Trick

Since only water is added, the amount of acid remains constant.

Initial Concentration × Initial Volume = Final Concentration × Final Volume

20% × 80 L = 10% × (80 + Added Water)

16 = 0.1 × (80 + Added Water) ⇒ 160 = 80 + Added Water ⇒ Added Water = 80 L.

∴ The correct answer is 80 L.

Alternate Method

Given: Initial volume = 80 L, Initial acid = 20%, Final acid = 10%, Water acid concentration = 0%.

We use the Alligation Method to find the ratio of Initial Solution to Added Water.

⇒ Ratio of Acid Solution to Water = 10 : 10 = 1 : 1.

⇒ Since 1 unit represents the Initial Volume = 80 L.

⇒ Therefore, 1 unit of Added Water = 80 L.

∴ The correct answer is 80 L.

Additional Information

Rule of Constant Component

In dilution problems, identify the substance whose quantity does not change. Use the formula: Initial % × Initial Total = Final % × Final Total.

Alligation Cross Method

Ratio of ingredients = (Higher Value − Mean Value) : (Mean Value − Lower Value). It is the fastest way to solve mixture problems involving two components.

Water Concentration in Mixtures

When pure water is added to an acid/milk solution, the concentration of acid/milk in the added water is always taken as 0%.

Shortcut Trick

Angle between tangents (∠APB) = 180° − (2 × Angle on major arc)

∠APB = 180° − (2 × 70°)

∠APB = 180° − 140°

∠APB = 40°

∴ The correct answer is 40°.

Alternate Method

Given: ∠ACB = 70° (where C is a point on the major arc AB).

Formula Used:

1. Angle subtended by a chord at the center is twice the angle on the circumference: ∠AOB = 2 × ∠ACB.
2. The angle between two tangents from an external point and the angle subtended by the radii at the center are supplementary: ∠APB + ∠AOB = 180°.

⇒ Angle at center ∠AOB = 2 × ∠ACB

⇒ ∠AOB = 2 × 70° = 140°

⇒ In quadrilateral OAPB, ∠OAP = 90° and ∠OBP = 90° (Radius ⊥ Tangent)

⇒ ∠APB + ∠OAP + ∠AOB + ∠OBP = 360°

⇒ ∠APB + 90° + 140° + 90° = 360°

⇒ ∠APB + 320° = 360°

⇒ ∠APB = 360° − 320° = 40°

∴ The correct answer is 40°.

Additional Information

Angle at Circumference Property

The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle (∠Center = 2 × ∠Circumference).

Tangent-Radius Theorem

A tangent to a circle is always perpendicular to the radius at the point of contact, making the angle 90°.

Cyclic Properties and Tangents

The external angle between two tangents is supplementary to the central angle formed by the radii at the points of contact (∠P + ∠O = 180°).

Shortcut Trick

Base Area (Trapezium) = 1/2 × (sum of parallel sides) × base height

Base Area = 1/2 × (4 + 8) × 5 = 6 × 5 = 30 cm²

Volume of Prism = Base Area × Prism Height

Volume = 30 × 10 = 300 cm³

∴ The correct answer is 300 cm³.

Alternate Method

Given:

Parallel sides of the trapezoidal base (a, b) = 4 cm and 8 cm

Height of the trapezoidal base (h) = 5 cm

Height of the right prism (H) = 10 cm

Formula Used:

Volume = Area of Base × Height of Prism

Area of Trapezium = 1/2 × (sum of parallel sides) × height

Calculations:

⇒ Area of base = 1/2 × (4 + 8) × 5

⇒ Area of base = 1/2 × 12 × 5

⇒ Area of base = 6 × 5 = 30 cm²

⇒ Volume of prism = 30 × 10

⇒ Volume of prism = 300 cm³

∴ The correct answer is 300 cm³.

Additional Information

Volume of a Prism

The volume of any prism is calculated as the Base Area × Height, regardless of the shape of the base.

Total Surface Area of a Prism

TSA = (Perimeter of base × Height) + 2 × Area of base.

Right Prism vs Oblique Prism

In a right prism, the side faces are perpendicular to the base, while in an oblique prism, they are not.

Shortcut Trick

In a right prism, the Lateral Surface Area (LSA) is directly calculated as Perimeter of Base × Height.

Given LSA = 300 cm² and Height = 5 cm.

Perimeter = LSA ÷ Height

Perimeter = 300 ÷ 5 = 60 cm.

∴ The correct answer is 60 cm.

Alternate Method

Given:

Base Area = 120 cm²

Height (h) = 5 cm

Lateral Surface Area (LSA) = 300 cm²

Formula Used:

Lateral Surface Area = Perimeter of the Base × Height

Calculations:

⇒ LSA = Perimeter × h

⇒ 300 = Perimeter × 5

⇒ Perimeter = 300 ÷ 5

⇒ Perimeter = 60 cm

∴ The correct answer is 60 cm.

Additional Information

Total Surface Area (TSA) of a Prism

TSA = Lateral Surface Area + 2 × Base Area. In this case, TSA = 300 + 2 × 120 = 540 cm².

Volume of a Prism

Volume = Base Area × Height. In this case, Volume = 120 × 5 = 600 cm³.

Right Prism Definition

A right prism is a 3D shape where the joining edges and faces are perpendicular to the base faces.

Shortcut Trick

Since ∠ADE = ∠ABC and ∠A is common, ΔADE ~ ΔABC by AA similarity.

Side ratio property: AD / AB = AE / AC

8 / 20 = AE / 25 ⇒ AE = (8 × 25) ÷ 20 = 10 cm.

Length of CE = AC − AE = 25 − 10 = 15 cm.

∴ The correct answer is 15 cm.

Alternate Method

Given: AD = 8 cm, AB = 20 cm, AC = 25 cm, ∠ADE = ∠ABC.

Formula Used: AA Similarity Criteria and ratio of corresponding sides.

In ΔADE and ΔABC:

⇒ ∠ADE = ∠ABC (Given)

⇒ ∠DAE = ∠BAC (Common angle)

⇒ ΔADE ~ ΔABC (By AA Similarity)

Corresponding sides are proportional:

⇒ AD / AB = AE / AC

⇒ 8 / 20 = AE / 25

⇒ 2 / 5 = AE / 25

⇒ AE = (2 × 25) ÷ 5 = 10 cm

Now, from the figure:

⇒ CE = AC − AE

⇒ CE = 25 − 10

⇒ CE = 15 cm

∴ The correct answer is 15 cm.

Additional Information

AA Similarity Rule

Two triangles are similar if two angles of one triangle are equal to two corresponding angles of the other triangle.

Side Proportionality

For similar triangles, the ratio of any two corresponding sides is always constant: AD/AB = AE/AC = DE/BC.

Basic Proportionality Theorem (BPT)

If a line is drawn parallel to one side of a triangle to intersect the other two sides, the other two sides are divided in the same ratio.

Shortcut Trick

Sum of ratio parts = 7 + 8 + 9 = 24 units.

Given, 24 units = Rs. 8,200.

C's share (9 units) = (8,200 ÷ 24) × 9.

C's share = (8,200 ÷ 8) × 3 = 1025 × 3 = Rs. 3,075.

∴ The correct answer is Rs. 3075.

Alternate Method

Given: Ratio of shares A : B : C = 7 : 8 : 9 and Total Sum = Rs. 8,200.

Let the shares of A, B, and C be 7x, 8x, and 9x respectively.

⇒ Total amount = 7x + 8x + 9x = 24x

⇒ 24x = 8,200

⇒ x = 8,200 ÷ 24 = 1025 ÷ 3

⇒ C's share = 9x

⇒ C's share = 9 × (1025 ÷ 3)

⇒ C's share = 3 × 1025 = 3,075

∴ The correct answer is Rs. 3075.

Additional Information

Ratio Distribution Rule

To divide a quantity S in the ratio a:b:c, the share of the third person is calculated as S × [c ÷ (a + b + c)].

Mean Proportional

The mean proportional between two numbers a and b is defined as √(a × b).

Third Proportional

If three numbers a, b, and c are in continued proportion (a:b = b:c), then c is the third proportional and c = b² ÷ a.

Shortcut Trick

Profit Ratio of Mahesh : Suresh = (75,000 × 12) : (90,000 × 4)

Ratio = (75 × 12) : (90 × 4) = 900 : 360 = 5 : 2

Total Ratio Parts = 5 + 2 = 7 units

7 units = ₹84,000 → 1 unit = ₹12,000

Mahesh's share (5 units) = 5 × 12,000 = ₹60,000

∴ The correct answer is ₹60,000.

Alternate Method

Given:

Mahesh's Investment (C₁) = ₹75,000

Mahesh's Investment Time (T₁) = 12 months

Suresh's Investment (C₂) = ₹90,000

Suresh's Investment Time (T₂) = 12 − 8 = 4 months

Total Profit = ₹84,000

Formula Used:

Profit Ratio = (Capital₁ × Time₁) : (Capital₂ × Time₂)

Calculations:

⇒ Mahesh's Profit : Suresh's Profit = (75,000 × 12) : (90,000 × 4)

⇒ Mahesh's Profit : Suresh's Profit = 9,00,000 : 3,60,000

⇒ Dividing by 1,80,000 ⇒ 5 : 2

⇒ Total Ratio = 5 + 2 = 7

⇒ Mahesh's share = (5 ÷ 7) × 84,000

⇒ Mahesh's share = 5 × 12,000

⇒ Mahesh's share = 60,000

∴ The correct answer is ₹60,000.

Additional Information

Partnership Profit Sharing Rule

When investments are for different time periods, profit is distributed in the ratio of the product of Capital and Time (P₁ : P₂ = C₁T₁ : C₂T₂).

Active vs. Sleeping Partner

An Active Partner manages the business and may receive a salary or extra share, while a Sleeping Partner only provides capital and shares profit/loss based on investment ratio.

Equivalent Capital for a Month

The product (Capital × Time) is also known as the Monthly Equivalent Capital, representing how much capital for 1 month would yield the same profit.

Shortcut Trick

Investment Ratio of Amit : Bhupendra : Chameli = 30,000 : 40,000 : 70,000 = 3 : 4 : 7

Total ratio units = 3 + 4 + 7 = 14 units

14 units = ₹ 56,000 (Total Profit)

1 unit = 56,000 ÷ 14 = ₹ 4,000

Bhupendra's share (4 units) = 4 × 4,000 = ₹ 16,000

∴ The correct answer is ₹ 16,000.

Alternate Method

Given:

Investment of Amit = ₹ 30,000

Investment of Bhupendra = ₹ 40,000

Investment of Chameli = ₹ 70,000

Total Profit = ₹ 56,000

Formula Used:

Profit Share ∝ Investment × Time (Since time is constant, Profit Ratio = Investment Ratio)

Calculations:

⇒ Ratio of profits = 30,000 : 40,000 : 70,000

⇒ Ratio = 3 : 4 : 7

⇒ Sum of ratio parts = 3 + 4 + 7 = 14

⇒ Bhupendra's share = (Ratio of Bhupendra ÷ Total Ratio) × Total Profit

⇒ Bhupendra's share = (4 ÷ 14) × 56,000

⇒ Bhupendra's share = 4 × (56,000 ÷ 14)

⇒ Bhupendra's share = 4 × 4,000

⇒ Bhupendra's share = ₹ 16,000

∴ The correct answer is ₹ 16,000.

Additional Information

Partnership Profit Rule

Profit is always distributed in the ratio of the product of Capital and Time: P₁ : P₂ = (C₁ × T₁) : (C₂ × T₂).

Active vs Sleeping Partner

An Active partner manages the business and may receive a fixed salary or percentage of profit before the remaining profit is divided among all partners.

Compound Partnership

If partners invest money for different time periods, it is called a Compound Partnership, and the time factor must be multiplied by the capital.

Shortcut Trick

Volume of Cone = (1/3) × π × r² × h

Direct substitution: 4620 = (1/3) × (22/7) × 21 × 21 × h

4620 = 22 × 21 × h

4620 = 462 × h

h = 4620 ÷ 462 = 10 cm

∴ The correct answer is 10 cm.

Alternate Method

Given:

Volume (V) = 4,620 cm³

Radius (r) = 21 cm

π = 22/7

Formula Used:

Volume of a right circular cone = (1/3) × π × r² × h

Calculations:

⇒ 4620 = (1/3) × (22/7) × 21 × 21 × h

⇒ 4620 = (1/3) × 22 × 3 × 21 × h

⇒ 4620 = 22 × 21 × h

⇒ 4620 = 462 × h

⇒ h = 4620 ÷ 462

⇒ h = 10 cm

∴ The correct answer is 10 cm.

Additional Information

Slant Height of a Cone

The slant height (l) is the distance from the apex to any point on the base circle: l = √(r² + h²).

Curved Surface Area (CSA)

The area of the curved surface of the cone is given by: CSA = π × r × l.

Total Surface Area (TSA)

The sum of the area of the base and the curved surface: TSA = π × r × (l + r).

Shortcut Trick

Radius (r) = Total Distance ÷ (2 × π × Number of Rounds)

Given Total Distance = 1000 m and Rounds = 4.

r = 1000 ÷ (2 × 3.14159 × 4) = 1000 ÷ 25.1327 ≈ 39.77 m

∴ The correct answer is 39.77 m.

Alternate Method

Given:

Total distance to be covered = 1 km = 1000 m

Total number of rounds = 4

Formula Used:

Total Distance = Number of rounds × Circumference of the track

Circumference = 2 × π × Radius (r)

Calculations:

⇒ Distance covered in 1 round = 1000 ÷ 4

⇒ Circumference = 250 m

⇒ 2 × (22/7) × r = 250

⇒ (44/7) × r = 250

⇒ r = (250 × 7) ÷ 44

⇒ r = 1750 ÷ 44

⇒ r = 39.7727... m

⇒ r ≈ 39.77 m

∴ The correct answer is 39.77 m.

Additional Information

Circumference of a Circle

The total boundary length of a circle is called its circumference, calculated as C = 2 × π × r, where π ≈ 3.14159 or 22/7.

Area of a Circle

The space enclosed within the boundary of a circle is its area, given by the formula A = π × r².

Unit Conversion

In racing track problems, always convert kilometers to meters (1 km = 1000 m) to maintain consistency with the radius in meters.